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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2601.02696 |
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| _version_ | 1866915709758472192 |
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| author | Conner, Gregory Kent, Curtis Luo, Jun Yang, Yi |
| author_facet | Conner, Gregory Kent, Curtis Luo, Jun Yang, Yi |
| contents | Let $λ_K:\bbR^2\rightarrow\{0,1,\ldots\}\cup\{\infty\}$ be the lambda function of a planar comapctum $K$, as defined in MR4488162. It is known that a planar continuum is locally connected if and only if its lambda function vanishes everywhere, or equivalently, $λ_K(K)=\{0\}$. In this article we show that every fractal square $K$ satisfies $λ_K(K)\subset\{0,1\}$ and find criterions to classify when $λ_K(K)$ equals $\{0\}$, $\{1\}$ or $\{0,1\}$. Here for any integer $N\ge2$ and any set $\Dc=\left\{(i,j): 0\le i,j\le N-1\right\}$ with cardinality $\ge2$, if we set $K^{(0)}=[0,1]^2$ and $\displaystyle K^{(n)}=\left\{\frac{x+d}{N}: x\in K^{(n-1)}, d\in\Dc\right\}(n\ge1)$ then $K=\bigcap_nK^{(n)}$ is called a fractal square. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02696 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Classification of Fractal Squares Conner, Gregory Kent, Curtis Luo, Jun Yang, Yi General Topology Let $λ_K:\bbR^2\rightarrow\{0,1,\ldots\}\cup\{\infty\}$ be the lambda function of a planar comapctum $K$, as defined in MR4488162. It is known that a planar continuum is locally connected if and only if its lambda function vanishes everywhere, or equivalently, $λ_K(K)=\{0\}$. In this article we show that every fractal square $K$ satisfies $λ_K(K)\subset\{0,1\}$ and find criterions to classify when $λ_K(K)$ equals $\{0\}$, $\{1\}$ or $\{0,1\}$. Here for any integer $N\ge2$ and any set $\Dc=\left\{(i,j): 0\le i,j\le N-1\right\}$ with cardinality $\ge2$, if we set $K^{(0)}=[0,1]^2$ and $\displaystyle K^{(n)}=\left\{\frac{x+d}{N}: x\in K^{(n-1)}, d\in\Dc\right\}(n\ge1)$ then $K=\bigcap_nK^{(n)}$ is called a fractal square. |
| title | A Classification of Fractal Squares |
| topic | General Topology |
| url | https://arxiv.org/abs/2601.02696 |